Maximization of quadratic forms expressed by distance matrices

Saichi Izumino; Noboru Nakamura

doi:10.14492/hokmj/1285766422

Maximization of quadratic forms expressed by distance matrices

Research output: Contribution to journal › Article

Abstract

If x, y, z are real numbers satisfying x + y + z = 1, then the maximum of the quadratic form axy + bxz + cyz with positive constants a, b, c is abc/2ab + 2ac + 2bc − a² − b² − c² under the assumption √a < √b + √c. Extending this fact, we give the maximum of the quadratic form ∑1≤i<j≤n a_ijx_ix_j in n-variables x₁, …, x_n satisfying ∑ⁿ _i=1 x_i = 1 with constants a_ij = 0 under certain assumptions.

Original language	English
Pages (from-to)	641-658
Number of pages	18
Journal	Hokkaido Mathematical Journal
Volume	35
Issue number	3
DOIs	https://doi.org/10.14492/hokmj/1285766422
Publication status	Published - Jan 1 2006
Externally published	Yes

Keywords

Distance matrix
Ozeki’s inequality
Quadratic form

ASJC Scopus subject areas

Mathematics(all)

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abstract = "If x, y, z are real numbers satisfying x + y + z = 1, then the maximum of the quadratic form axy + bxz + cyz with positive constants a, b, c is abc/2ab + 2ac + 2bc − a2 − b2 − c2 under the assumption √a < √b + √c. Extending this fact, we give the maximum of the quadratic form ∑1≤iijxixj in n-variables x1, …, xn satisfying ∑n i=1 xi = 1 with constants aij = 0 under certain assumptions.",

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AU - Nakamura, Noboru

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AB - If x, y, z are real numbers satisfying x + y + z = 1, then the maximum of the quadratic form axy + bxz + cyz with positive constants a, b, c is abc/2ab + 2ac + 2bc − a2 − b2 − c2 under the assumption √a < √b + √c. Extending this fact, we give the maximum of the quadratic form ∑1≤iijxixj in n-variables x1, …, xn satisfying ∑n i=1 xi = 1 with constants aij = 0 under certain assumptions.

KW - Distance matrix

KW - Ozeki’s inequality

KW - Quadratic form

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